# Quantum Bayesian Networks

## January 4, 2010

### I Don’t Have to Show You Any Stinking Mixed States

Filed under: Uncategorized — rrtucci @ 2:41 pm

From the 1948 film “The Treasure of the Sierra Madre” with Humphrey Bogart

Dobbs: If you’re the police where are your badges?
Gold Hat: Badges? We ain’t got no badges. We don’t need no badges! I don’t have to show you any stinkin’ badges!

QB nets don’t need any stinking mixed states. Given a mixed state, you can always replace it by a pure state (called a “purification”) that lives in a bigger space. Technically, if a mixed state is described by a density matrix $\rho_1$ acting on a Hilbert space ${\cal H}_1$, you can always find a Hilbert space ${\cal H}_2$ and a pure state $|\Psi_{12}\rangle\in {\cal H}_1\otimes{\cal H}_2$ such that $\rho_1= tr_2(|\Psi_{12}\rangle\langle\Psi_{12}|)$. For example, if $\rho_1$ has eigenvalues $w_r$ and eigenvectors $|\psi_r\rangle$ so that $\rho_1 = \sum_r w_r |\psi_r\rangle\langle\psi_r|$, then a possible purification of $\rho_1$ is $|\Psi_{12}\rangle = \sum_r \sqrt{w_r} |\psi_r\rangle\otimes |\psi_r\rangle$.